In nature, substances often possess a crystalline structure. The basic component of a crystal is the unit cell, such as the rectangular parallelpiped illustrated in ( Figure 1). In the questions that follow, express your answers in terms of the unit vectors i, j, and Â, that is, a vector with components VI, Vy, and V₂ in the x, y, and z directions, respectively, is written V₁₁ + V₁₁+V₂k. Part C What is the vector VoF from point O to point F? Express you answer in terms of any of the following: L1, L2, L3, i, j, and k. VOF= ΜΕ ΑΣΦ ? Figure L₁ E H A L₁ F B 0 0 1 of 1 Submit Request Answer Part D What is the vector from A to B, VAB? Express you answer in terms of any of the following: L1, L2, L3, i, j, and k. VAB = ΜΕ ΑΣΦ ? Submit Request Answer

Figure 1.19 shows two vectors lying in the xy-plane. Determine the signs of the x- and y-components of A, B, and A+B.

A pirate has buried his treasure on an island with five trees located at the points (30.0 m, 20.0 m), (60.0 m, 80.0 m), (10.0 m, 10.0 m), (40.0 m, 30.0 m), and (70.0 m, 60.0 m), all measured relative to some origin, as shown in Figure P1.69. His ships log instructs you to start at tree A and move toward tree B, but to cover only one-half the distance between A and B. Then move toward tree C, covering one-third the distance between your current location and C. Next move toward tree D, covering one-fourth the distance between where you are and D. Finally move toward tree E, covering one-fifth the distance between you and E, stop, and dig. (a) Assume you have correctly determined the order in which the pirate labeled the trees as A, B, C, D, and E as shown in the figure. What are the coordinates of the point where his treasure is buried? (b) What If? What if you do not really know the way the pirate labeled the trees? What would happen to the answer if you rearranged the order of the trees, for instance, to B (30 m, 20 m), A (60 m, 80 m), E (10 m, 10 m), C (40 m, 30 m), and D (70 m, 60 m)? State reasoning to show that the answer does not depend on the order in which the trees are labeled. Figure 1.69

For what values of a are the vectors A = 2ai − 2j + ak and B = ai + 2aj + 2k perpendicular?